Finite Group Databasebeta

← C2×#11749408 #14423529

Subgroups — C2×#11749408 #14423529

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6769C211yesmaximal, characteristic
2C22×#6717C211yesmaximal, characteristic, complemented
3C2×C5×#946C211yesmaximal, characteristic, complemented
4C22×C3×#689C211yesmaximal, characteristic, complemented
5#11749408C218yesmaximal, direct product factor
6C2×#90970C211yesmaximal, characteristic, complemented
7C2×#90969C211yesmaximal, characteristic, complemented
8C2×(C2×D3)⋊#168C211yesmaximal, characteristic, complemented
9C2×#65336—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×D60C2211yescharacteristic
12C22×C4×C5×D3C2211yescharacteristic
13C22×C3×Dic10C2211yescharacteristic
14C22×#1203—21no—
15C22×#1200—21no—
16C2×#6769—24no—
17C2×#6769—22no—
18C2×#6769—24no—
19C2×#6769C2214yes—
20C2×#6717—24no—
21C2×#6717C2214yescomplemented
22C22×C3⋊#168C2211yescharacteristic
23C5×#946—22no—
24C5×#946C2214yescomplemented
25C2×C5×(C2×D3)⋊C4—21no—
26C2×C5×C4⋊Dic3C2211yescharacteristic
27C2×C5×Dic3⋊C4—21no—
28C2×C3×#689—24no—
29C2×C3×#689C2214yescomplemented
30C22×C3×#168C2211yescharacteristic
31#90970C2214yescomplemented
32#90969C2214yescomplemented
33(C2×D3)⋊#168C2214yescomplemented
34C2×#5187—51nomaximal, Hall
35C22×#709—31no—
36C2×C5×C22⋊Q8—31no—
37C22×#689D311yescharacteristic, complemented
38C22×#688—31no—
39C2×C22⋊#168—31no—
40#65336—38no—
41C2×#3942—31no—
42C2×#3941—31no—
43C23×D15—21no—
44C23×C5×D3D411yescharacteristic
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8—22no—
47C2×C3×C5×Q8—22no—
48C2×C3×C5×Q8C2312yes—
49C22×C3×C4×C5—21no—
50C22×C3×C4×C5C2311yescharacteristic
51C2×D60—21no—
52C2×D60—22no—
53C2×D60D414yes—
54C2×D60C2312yes—
55C22×C5×Dic3—21no—
56C22×C5×Dic3D411yescharacteristic
57C2×C4×C5×D3—22no—
58C2×C4×C5×D3—22no—
59C2×C4×C5×D3D412yes—
60C2×C4×C5×D3C2312yes—
61C22×C3×Dic5—21no—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—22no—
64C2×C3×Dic10D414yes—
65C2×C3×Dic10C2312yes—
66C2×#1203—28no—
67C2×#1203—24no—
68C2×#1200—28no—
69C2×#1200—24no—
70#6769—216no—
71#6769—28no—
72#6769—28no—
73#6717—28no—
74C2×C3⋊#168—22no—
75C2×C3⋊#168C2312yes—
76C5×(C2×D3)⋊C4—24no—
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2312yes—
79C5×Dic3⋊C4—24no—
80C3×#689—28no—
81C2×C3×#168—22no—
82C2×C3×#168C2312yes—
83C22×#239—51no—
84C22×C3×Q16—51no—
85C2×#946D511yescharacteristic, complemented
86C22×#215—51no—
87C2×#835—51no—
88#5187—58no—
89C2×#821—51no—
90C2×(C2×D3)⋊C8—51no—
91C22×C5×Q8C2×D311yescharacteristic
92C23×C4×C5—31no—
93C22×C22⋊D5—61no—
94C2×#709—34no—
95C2×#709—64no—
96C2×#709—62no—
97C2×#709—64no—
98C22×D20—31no—
99C22×C4×D5—61no—
100C22×Dic10C2×D311yescharacteristic
101C5×C22⋊Q8—34no—
102C5×C22⋊Q8—62no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4—61no—
105C2×C5×C22⋊C4—61no—
106C2×#689—24no—
107C2×#689C2×D314yescomplemented
108C2×#688—34no—
109C2×#688—64no—
110C22×#168—31no—
111C22×#168C2×D311yescharacteristic
112C22⋊#168—34no—
113#3942—34no—
114#3941—34no—
115C2×#452—151noSylow
116C23×C3×C5C2×D411yescharacteristic
117C22×D15—24no—
118C22×D15—22no—
119C22×D15—42no—
120C22×D15—42no—
121C22×C5×D3—22no—
122C22×C5×D3—22no—
123C22×C5×D3—22no—
124C22×C5×D3C2×D412yes—
125C3×C5×Q8—22no—
126C3×C5×Q8—22no—
127C3×C5×Q8—42no—
128C2×C3×C4×C5—22no—
129C2×C3×C4×C5—42no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D412yes—
132C2×C3×C4×C5C2×D412yes—
133C2×C3×C4×C5C2411yescharacteristic
134D60—24no—
135D60—22no—
136D60C2×D414yes—
137C2×C5×Dic3—22no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—42no—
140C2×C5×Dic3C2×D412yes—
141C4×C5×D3—24no—
142C4×C5×D3—22no—
143C4×C5×D3—22no—
144C2×C3×Dic5—24no—
145C2×C3×Dic5—22no—
146C3×Dic10—24no—
147C3×Dic10—22no—
148C3×Dic10C2×D414yes—
149#1203—216no—
150#1200—216no—
151C3⋊#168—22no—
152C3×#168—22no—
153C22×C3×Q8—51no—
154C22×C3×Q8C2×D511yescharacteristic
155C2×#239—54no—
156C2×#239—104no—
157C2×#239—104no—
158C2×#239—102no—
159C22×D12—51no—
160C22×D12—101no—
161C22×C4×D3—101no—
162C22×C4×D3C2×D511yescharacteristic
163C2×C3×Q16—54no—
164C2×C3×Q16—104no—
165C22×C3×C8—51no—
166#946—22no—
167#946C2×D514yescomplemented
168C2×#215—54no—
169C2×#215—104no—
170C2×(C2×D3)⋊C4—21no—
171C2×C4⋊Dic3C2×D511yescharacteristic
172C2×Dic3⋊C4—21no—
173C22×C3⋊C8—51no—
174#835—54no—
175#821—54no—
176(C2×D3)⋊C8—54no—
177C24×C5—31no—
178C23×D5—61no—
179C2×C5×Q8—21no—
180C2×C5×Q8—22no—
181C2×C5×Q8—22no—
182C2×C5×Q8C22×D312yes—
183C22×C4×C5—21no—
184C22×C4×C5—31no—
185C22×C4×C5—32no—
186C22×C4×C5—32no—
187C22×C4×C5—62no—
188C22×C4×C5—62no—
189C22×C4×C5—61no—
190C22×C4×C5C22×D311yescharacteristic
191C2×C22⋊D5—64no—
192C2×C22⋊D5—68no—
193C22×Dic5—21no—
194#709—68no—
195#709—616no—
196#709—68no—
197C2×D20—34no—
198C2×D20—32no—
199C2×D20—61no—
200C2×D20—62no—
201C2×C4×D5—64no—
202C2×C4×D5—68no—
203C2×Dic10—21no—
204C2×Dic10—22no—
205C2×Dic10#6714yes—
206C2×Dic10C22×D312yes—
207C5×C4⋊C4—32no—
208C5×C4⋊C4—64no—
209C5×C4⋊C4—61no—
210C5×C22⋊C4—64no—
211#689—28no—
212#688—68no—
213C2×#168—22no—
214C2×#168—32no—
215C2×#168—62no—
216C2×#168C22×D312yes—
217C22×#41—151no—
218C22×Q16—51no—
219C22×SD16—151no—
220C2×C22⋊Q8—31no—
221#452—158no—
222C2×#103—151no—
223C2×#102—151no—
224C2×C22⋊C8—151no—
225C22×C3×C5#12012yes—
226C22×C3×C5#13512yes—
227C22×C3×C5#13712yes—
228C22×C3×C5C22×D411yescharacteristic
229C2×D15—24no—
230C2×D15—42no—
231C2×D15—42no—
232C2×D15—48no—
233C2×C5×D3—22no—
234C2×C5×D3—22no—
235C2×C5×D3—22no—
236C2×C5×D3—44no—
237C3×C4×C5—42no—
238C3×C4×C5#12012yes—
239C3×C4×C5C22×D411yescharacteristic
240C3×C4×C5C22×D411yescharacteristic
241C3×Dic5—24no—
242C5×Dic3—22no—
243C5×Dic3—42no—
244C23×D3—101no—
245C23×D3C22⋊D511yescharacteristic
246C2×C3×Q8—21no—
247C2×C3×Q8—22no—
248C2×C3×Q8—22no—
249C2×C3×Q8—54no—
250C2×C3×Q8—52no—
251C2×C3×Q8—102no—
252C2×C3×Q8—101no—
253C2×C3×Q8C22×D512yes—
254C22×C3×C4—21no—
255C22×C3×C4—101no—
256C22×C3×C4C22×D511yescharacteristic
257C22×Dic3—21no—
258C22×Dic3C22⋊D511yescharacteristic
259#239—108no—
260#239—108no—
261#239—1016no—
262C2×D12—54no—
263C2×D12—52no—
264C2×D12—108no—
265C2×D12—104no—
266C2×D12—102no—
267C2×D12—101no—
268C2×C4×D3—22no—
269C2×C4×D3—22no—
270C2×C4×D3—108no—
271C2×C4×D3—104no—
272C2×C4×D3C22⋊D512yes—
273C2×C4×D3C22×D512yes—
274C3×Q16—108no—
275C2×C3×C8—52no—
276C2×C3×C8—102no—
277#215—108no—
278(C2×D3)⋊C4—24no—
279C4⋊Dic3—21no—
280C4⋊Dic3C22×D512yes—
281Dic3⋊C4—24no—
282C2×C3⋊C8—52no—
283C2×C3⋊C8—102no—
284C23×C5—32no—
285C23×C5—62no—
286C23×C5—62no—
287C23×C5—62no—
288C23×C5D3×D411yescharacteristic
289C22×D5—64no—
290C22×D5—62no—
291C22×D5—122no—
292C22×D5—122no—
293C5×Q8—22no—
294C5×Q8—22no—
295C5×Q8—42no—
296C2×C4×C5—22no—
297C2×C4×C5—32no—
298C2×C4×C5—42no—
299C2×C4×C5—62no—
300C2×C4×C5—64no—
301C2×C4×C5—62no—
302C2×C4×C5—62no—
303C2×C4×C5—62no—
304C2×C4×C5—122no—
305C2×C4×C5D3×D411yescharacteristic
306C2×C4×C5D3×D412yes—
307C2×C4×C5C2×#6712yes—
308C2×C4×C5C23×D311yescharacteristic
309C22⋊D5—616no—
310C2×Dic5—24no—
311C2×Dic5—22no—
312D20—34no—
313D20—64no—
314D20—62no—
315C4×D5—616no—
316Dic10—24no—
317Dic10—22no—
318Dic10C2×#6714yes—
319#168—22no—
320#168—62no—
321C2×#41—154no—
322C2×#41—302no—
323C2×#41—304no—
324C2×#41—304no—
325C22×Q8—51no—
326C22×Q8D3×D511yescharacteristic
327C22×D4—151no—
328C22×D4—301no—
329C23×C4—31no—
330C23×C4—301no—
331C2×Q16—54no—
332C2×Q16—104no—
333C2×SD16—154no—
334C2×SD16—304no—
335C22×C8—51no—
336C22×C8—151no—
337C22⋊Q8—34no—
338C22⋊Q8—62no—
339C2×C4⋊C4—31no—
340C2×C4⋊C4—61no—
341C2×C22⋊C4—61no—
342#103—154no—
343#102—154no—
344C22⋊C8—154no—
345C2×C3×C5#45214yescomplemented
346C2×C3×C5C2×#12011yescharacteristic
347C2×C3×C5C2×#13511yescharacteristic
348C2×C3×C5C2×#13711yescharacteristic
349D15—44no—
350C5×D3—42no—
351C23×C3C2×C22⋊D511yescharacteristic
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—102no—
356C22×D3—104no—
357C22×D3—202no—
358C22×D3—202no—
359C22×D3C2×C22⋊D512yes—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—42no—
363C3×Q8—54no—
364C3×Q8—102no—
365C3×Q8—104no—
366C2×C3×C4—22no—
367C2×C3×C4—42no—
368C2×C3×C4—102no—
369C2×C3×C4—104no—
370C2×C3×C4D4×D512yes—
371C2×C3×C4D4×D512yes—
372C2×C3×C4C2×C22⋊D511yescharacteristic
373C2×C3×C4C23×D511yescharacteristic
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—42no—
377C2×Dic3C2×C22⋊D512yes—
378D12—54no—
379D12—1016no—
380D12—102no—
381D12—104no—
382C4×D3—22no—
383C4×D3—22no—
384C4×D3—24no—
385C4×D3—1016no—
386C3×C8—102no—
387C3⋊C8—102no—
388C22×C5—62no—
389C22×C5—62no—
390C22×C5—62no—
391C22×C5—124no—
392C22×C5#91812yes—
393C22×C5#91912yes—
394C22×C5#93712yes—
395C22×C5C2×D3×D411yescharacteristic
396C2×D5—64no—
397C2×D5—128no—
398C2×D5—122no—
399C2×D5—122no—
400C4×C5—42no—
401C4×C5—62no—
402C4×C5—122no—
403C4×C5#93712yes—
404C4×C5C2×D3×D411yescharacteristic
405C4×C5C22×#6711yescharacteristic
406Dic5—24no—
407C24—31no—
408C24—301no—
409#41—308no—
410#41—308no—
411#41—3016no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—54no—
416C2×Q8—52no—
417C2×Q8—101no—
418C2×Q8—102no—
419C2×Q8C2×D3×D512yes—
420C2×D4—154no—
421C2×D4—152no—
422C2×D4—304no—
423C2×D4—308no—
424C2×D4—302no—
425C2×D4—301no—
426C22×C4—21no—
427C22×C4—31no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—101no—
434C22×C4—304no—
435C22×C4—308no—
436C22×C4C2×D3×D511yescharacteristic
437Q16—108no—
438SD16—308no—
439C2×C8—52no—
440C2×C8—102no—
441C2×C8—152no—
442C2×C8—302no—
443C4⋊C4—32no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C22⋊C4—64no—
447C3×C5C2×#45211yescharacteristic, complemented, Hall
448C22×C3#408312yes—
449C22×C3#408812yes—
450C22×C3#409612yes—
451C22×C3C22×C22⋊D511yescharacteristic
452C2×D3—22no—
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—44no—
456C2×D3—104no—
457C2×D3—208no—
458C2×D3—202no—
459C2×D3—202no—
460C3×C4—42no—
461C3×C4—104no—
462C3×C4#408312yes—
463C3×C4C2×D4×D511yescharacteristic
464C3×C4C2×D4×D511yescharacteristic
465Dic3—22no—
466Dic3—42no—
467C2×C5—122no—
468C2×C5#518714yescomplemented
469C2×C5C2×#91811yescharacteristic
470C2×C5C2×#91911yescharacteristic
471C2×C5C2×#93711yescharacteristic
472D5—124no—
473C23—32no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—302no—
478C23—304no—
479C23—602no—
480C23—602no—
481C23D3×C22⋊D511yescharacteristic
482Q8—22no—
483Q8—22no—
484Q8—42no—
485Q8—54no—
486Q8—102no—
487Q8—104no—
488D4—154no—
489D4—3016no—
490D4—302no—
491D4—304no—
492C2×C4—22no—
493C2×C4—32no—
494C2×C4—42no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—102no—
501C2×C4—104no—
502C2×C4—122no—
503C2×C4—3016no—
504C2×C4D5×#6712yes—
505C2×C4D3×C22⋊D511yescharacteristic
506C2×C4#679012yes—
507C2×C4C22×D3×D511yescharacteristic
508C8—102no—
509C8—302no—
510C2×C3#6533614yescomplemented
511C2×C3C2×#408311yescharacteristic
512C2×C3C2×#408811yescharacteristic
513C2×C3C2×#409611yescharacteristic
514D3—42no—
515D3—204no—
516C5C2×#518711yescharacteristic, complemented, Sylow
517C22—62no—
518C22—62no—
519C22—62no—
520C22—124no—
521C22—304no—
522C22—608no—
523C22—602no—
524C22—602no—
525C22#9151912yes—
526C22#9152012yes—
527C22#9157712yes—
528C22C2×D3×C22⋊D511yescharacteristic
529C4—42no—
530C4—62no—
531C4—104no—
532C4—122no—
533C4#9157712yes—
534C4C2×D5×#6711yescharacteristic
535C4C2×#679011yescharacteristic
536C3C2×#6533611yescharacteristic, complemented, Sylow
537C2—122no—
538C2—604no—
539C2#1174940814yesdirect product factor
540C2C2×#9151911yescharacteristic
541C2C2×#9152011yescharacteristic
542C2C2×#9157711yescharacteristic
subgroupG/H
centerC23D3×C22⋊D5
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
prime radicalC22×#6769C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#452
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#6717C2semidirect
3. C2×C5×#946C2semidirect
4. C22×C3×#689C2semidirect
5. #11749408C2direct (N × H)
6. C2×#90970C2semidirect
7. C2×#90969C2semidirect
8. C2×(C2×D3)⋊#168C2semidirect
21. C2×#6717C22semidirect
24. C5×#946C22semidirect
29. C2×C3×#689C22semidirect
31. #90970C22semidirect
32. #90969C22semidirect
33. (C2×D3)⋊#168C22semidirect
37. C22×#689D3semidirect
85. C2×#946D5semidirect
107. C2×#689C2×D3semidirect
167. #946C2×D5semidirect
345. C2×C3×C5#452semidirect
447. C3×C5C2×#452semidirect
468. C2×C5#5187semidirect
510. C2×C3#65336semidirect
516. C5C2×#5187semidirect
536. C3C2×#65336semidirect
539. C2#11749408direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#67692
C22×#67172
C2×C5×#9462
C22×C3×#6892
#117494082
C2×#909702
C2×#909692
C2×(C2×D3)⋊#1682
C2×#653363
C2×#51875